Analysis of the Influence of the Vortex Shedder Shape on the Metrological

sensors

Article Analysis of the Inﬂuence of the Vortex Shedder Shape on the Metrological Properties of the Vortex Flow Meter

Mariusz R. Rzasa 1,* and Beata Czapla-Nielacna 2

1 Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, Proszkowska 76 Street, 45-758 Opole, Poland 2 Faculty of Mechanical Engineering, Opole University of Technology, Mikolajczyka 5 Street, 45-271 Opole, Poland; [email protected] * Correspondence: [email protected]; Tel.: +48-602-345-162

Abstract: Vortex flow meters are used to measure the flow of gases and liquids. The flow meters of this type measure the frequency of vortices that arise behind an obstacle set in the path of the flowing fluid. The frequency is a function of the speed of the flowing fluid. This obstacle is called the vortex shedder bar. The advantage of this solution is that the frequency of vortices does not viscose on the rheological properties of the fluid, such as viscosity or density. As a result, the indications of the vortex flowmeter do not depend on the temperature and type of fluid. The work includes numerical and experimental studies of the effect of changing the shape of a vortex generator on the stability of vortex generation in a vortex flowmeter. The article presents a numerical analysis of the influence of selected surfaces of the vortex shedder on the parameters of the vortex flowmeter. In order to determine the influence of the shape of the vortex shedder on the type of generated vortices,

simulations were carried out for various ﬂow velocities. Numerical calculations were experimentally veriﬁed for a cylinder-shaped vortex shedder. The experimental tests consist in determining the

Citation: Rzasa, M.R.; velocity field behind the vortex shedder. For this purpose, a proprietary method of determining local Czapla-Nielacna, B. Analysis of the liquid velocities and the visualization of local vortices were used. On the basis of the conducted Influence of the Vortex Shedder Shape research, the influence of the shape of the vortex shedder on the width of the von Karman vortex on the Metrological Properties of the street was determined and the optimal longitudinal distance from the shedder was determined in Vortex Flow Meter. Sensors 2021, 21, which it is most useful to measure the frequency of the vortices. This place ensures the stability of the 4697. https://doi.org/10.3390/ frequency of the generated vortices. s21144697 Keywords: vortex flow meter; flow visualization; flow measurement; flow visualization Academic Editor: Giovanni Betta

Received: 10 June 2021 Accepted: 7 July 2021 1. Introduction Published: 9 July 2021 One of the solutions used to measure the velocity of liquids are vortex flow meters. These flow meters are used to measure the velocity of liquids of various densities and Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in viscosities. Their advantage is that the measurement result is not significantly influenced by published maps and institutional affil- solid particles carried with the liquid. They can therefore be used to measure contaminated iations. liquids. A second advantage is the very low pressure drop across the measuring device. Their disadvantage is the relatively small measuring range [1]. The principle of operation of the vortex flowmeter is to create regular vortices behind the generator placed in the flowing liquid. The frequency of the vortices is a function of the flow velocity. The vortex generator can have different shapes. This shape has a direct Copyright: © 2021 by the authors. impact on the sensitivity and measuring range of the flowmeter. The frequency of the Licensee MDPI, Basel, Switzerland. This article is an open access article resulting vortices is a measure of the flow velocity, which is expressed by the Strouhal distributed under the terms and number [2]: f ·d conditions of the Creative Commons St = , (1) Attribution (CC BY) license (https:// v creativecommons.org/licenses/by/ where: St—Strouhal number, f —frequency of vortices, d—characteristic dimension, v— 4.0/). flow velocity.

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where: St—Strouhal number, f—frequency of vortices, d—characteristic dimension, v— flow velocity. The Strouhal number is constant regardless of the velocity of the liquid that passes The Strouhal number is constant regardless of the velocity of the liquid that passes the vortex generator. The advantage of this solution is that the frequency of the generated the vortex generator. The advantage of this solution is that the frequency of the generated von Karman vortices does not depend on the rheological properties of the fluid, such as von Karman vortices does not depend on the rheological properties of the fluid, such as viscosity or density. As a result, the indications of the vortex flowmeter do not depend on viscosity or density. As a result, the indications of the vortex flowmeter do not depend on the temperature and type of fluid. the temperature and type of fluid. It should be noted, however, that the von Karman vortex path arises for the turbulen It should be noted, however, that the von Karman vortex path arises for the turbulen flow. Thus, the Reynolds number should be greater than 5000 [3]. Therefore, the size of flow. Thus, the Reynolds number should be greater than 5000 [3]. Therefore, the size of the vortex shedder bar should be adapted to the rheological properties of the liquid and the vortex shedder bar should be adapted to the rheological properties of the liquid and the measuring range. The rheological properties of the fluid depend on the type of fluid, the measuringtemperature range. and The its physicochemical rheological properties composition. of the fluid Moreover, depend the on presence the type of of a significantfluid, temperatureamount and of solid its physicochemical particles in a liquid composition. significantly Moreover, changes the itspresence rheological of a significant properties [4]. amountWhen of solid selecting particles the diameterin a liquid of significantly the vortex shedder, changes the its valuerheological of the properties Reynolds [4]. number Whencalculated selecting fromthe diameter the formula of the should vortex be considered.shedder, the value of the Reynolds number calculated from the formula should be considered. v·dρ Re =⋅ , (2) 𝑅𝑒 = , (2) µ where:where: ρ—densityρ—density of the of fluid, the fluid, μ—dynamicµ—dynamic viscosity viscosity of the of fluid. the fluid. In thisIn work, this work, tests tests were were carried carried out out for for pure pure water water at at a a temperature temperature of 2020 ◦°CCin in the the speed speedrange range from from 0.5 0.5 to to 2 2 m/s. m/s. Assuming Assuming a a vo vortexrtex generator generator size size of of 20 20 mm, mm, Reynolds Reynolds num- numbers bers areare between between 10,000 10,000 and and 40,000. 40,000. Thus, Thus, this this is is sufficient sufficient to to produce produce a avon von Karman Karman street. street. FigureFigure 1 shows1 shows a typical a typical design design of a flowmeter of a flowmeter [5]. It [ 5consists]. It consists of a vortex of a vortex generator, generator, whichwhich is mounted is mounted in the inflow the space. flow It space. causes It vortices causes to vortices form behind to form it, behindcreating it, the creating char- the acteristiccharacteristic von Karman von vortex Karman path. vortex In order path. to In measure order to the measure frequency the frequency of the vortices of the that vortices arise,that a pressure arise, adetector pressure is detectormounted isat mounteda certain distance. at a certain The distance. simplest solution The simplest is a pres- solution sure sensoris a pressure with high sensor measurement with high measurementdynamics. The dynamics. frequency Theof the frequency vortices is of measured the vortices is electronically.measured electronically.

Vortex detector

Shedder bar

FigureFigure 1. Vortex 1. Vortex flowmeter. ﬂowmeter.

The shapeThe shape of the of vortex the vortex sheder sheder has a has signifi a significantcant impact impact on the on theparameters parameters and and met- metro- rologicallogical properties properties of the of theflow flow meter meter [6]. [Its6]. shape Its shape should should ensure ensure the formation the formation of regular of regular vorticesvortices in the in largest the largest possible possible measuring measuring range. range. The most The mostcommon common shape shape is a cylinder. is a cylinder. ThereThere are many are many works works on this on subject this subject in the in lit theerature literature [1,7]. [Classic1,7]. Classic designs designs of vortex of vortex flow flow metersmeters are also are used also usedin two-phase in two-phase flows flows [8]. [The8]. Theresearch research in this in this paper paper will will be becompared compared to to a cylinder-shapeda cylinder-shaped shedder shedder bar. bar. The The main main di disadvantagesadvantage of of the the cylindrical cylindrical shape shape is its very very narrow measuring rangerange [[9].9]. Therefore,Therefore, therethere is is a a need need to to look look for for other other solutions solutions [2 [2].]. There Thereare are many many works works in in the the literatureliterature whichwhich describedescribe cylindricalcylindrical shedder bar [1,7,10]. [1,7,10]. In In this this paper,paper, this shape willwill bebe usedused as as a a reference reference in in the the assessment assessment of theof the influence influence of selected of selectedshedder shedder surfaces surfaces on the on vonthe Karmanvon Karman vortex vortex path. path. The mainThe main disadvantage disadvantage of the of cylindrical the cylindricalshape shape is its narrow is its narrow measuring measuring range rang duee to due the to stability the stability of the of generated the generated vortices vor- [7]. As tices the[7]. sizeAs the and size shape and of shape the vortex of the generatorvortex generator has a large has impacta large onimpact the correct on the operationcorrect of operationthe vortex of the flowmetervortex flowmeter [2,11], the [2,11], search the forsearch new for solutions new solutions is still justifiedis still justified [9,12], [9,12], despite the fact that many shapes of vortex generators are known. despite the fact that many shapes of vortex generators are known. Many vortex shedder bars solutions are used in commercial flow meters. Many vortex generator solutions are used in commercial flow meters. Despite so many solutions, no

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Many vortex shedder bars solutions are used in commercial flow meters. Many vortex generator solutions are used in commercial flow meters. Despite so many solutions, nouniversal universal shape shape of theof the vortex vortex shedder shedder bar bar has has been been developed. developed. This This study study is devotedis devoted to tothe the analysis analysis of of the the influence influence of of selected selected surfaces surfaces of of the the vortex vortex sheddershedder barbar onon selectedselected parameters of the vortex flowme flowmeter.ter. In the firstfirst part, numericalnumerical tests were carried out, the aim of which was to distinguish significantsignificant su surfacesrfaces of the vortex generator, which may have a a significant significant impact impact on on the the metrological metrological parameters parameters of ofthe the flow flow meter. meter. Then, Then, the these- lectedselected shapes shapes were were tested tested experimentally. experimentally. A cylinder withwith aa diameterdiameter of ofd d= = 20 20 mm mm was was adopted adopted as as the the reference reference shape shape (Figure (Figure2 ). 2).Eight Eight surfaces surfaces were were distinguished distinguished in the in cross-section.the cross-section. Four Four of them of them form form the rakethe rake face face and andthe remainingthe remaining four four form form the trailingthe trailing face. face.

Figure 2. The basic vortex shedder shape.

This solution can be used to measure the fl flowow of clean and contaminated fluidsfluids with solid particles. Measurement error of the currentcurrent valuevalue ofof thethe streamstream 0.50.5÷ ÷ 1% for liquids and approx. approx. 2% 2% for for gases. gases. The The measurement measurement erro errorr is significantly is significantly influenced influenced by the by regu- the larityregularity of the of vortices the vortices formed. formed. The Thecomplex complex nature nature of the of theresulting resulting chips chips provides provides a lot a lotof informationof information about about the the process process of of their their formation. formation. This This information information can can be be processed on the basis of data fusing process [[13].13]. In order to determine the influenceinfluence of individual surfaces on thethe vonvon KarmanKarman swirlswirl pathway formed behind the vortex shedder bar.bar. The main analysis was the influenceinfluence of individual surfaces on the stability of the generated vortices. The results of the calculations were compared to the calculationscalculations for thethe cylinder-shapedcylinder-shaped generator.generator. The research was carried out in such a way that selected surfaces from 1 to 8 were modifiedmodified (Figure2 2)) andand then numerical calculations were were carried carried out out for for the the same same conditions conditions as as for for the the cylinder. cylinder. Shapes with a modifiedmodified rake face were proposed (Figure3 3a).a). TheThe secondsecond modificationmodification isis the changes in the runoffrunoff surfacesurface (Figure(Figure3 3b).b). TheThesame samecharacteristic characteristic dimensiondimension dd= = 2020 mm was kept in all modifications. modifications. In addition, the effect of extending the dimension of the sherdder bar in the direction of flow motion was tested (Figure3c). The elongation was sherdder bar in the direction of flow motion was tested (Figure 3c). The elongation was 6 6 mm (W1) and 12 mm (W2). In addition, it was checked what effect is the use of a slot mm (W1) and 12 mm (W2). In addition, it was checked what effect is the use of a slot with with a radius of 3 mm (W3) and a slot with a radius of 6 mm (W4) on the lower and upper a radius of 3 mm (W3) and a slot with a radius of 6 mm (W4) on the lower and upper surfaces of the shedder bar. surfaces of the shedder bar. The aim of the work is to present the influence of changes in the shape of the rake and runoff surface in the vortex shedder bar on the stability of the vortices. The work carried out numerical and experimental tests on the basis of which the parameters for various shapes of vortex shedder bar were determined.

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FigureFigure 3. 3.Shapes Shapes resulting resulting fromfrom (a) rake surface surface modification modiﬁcation (b (b) )runoff runoff surface surface modification modiﬁcation (c) (oval.c) oval.

2. NumericalThe aim Methodof the work is to present the influence of changes in the shape of the rake and Manyrunoff mathematicalsurface in the models vortex areshedder evaluated bar on to the simulate stability flow of velocity.the vortices. The mostThe work popular modelscarried forout modeling numerical complex and experimental flow problems tests areon numericalthe basis of methods which the CFD parameters (Computational for Fluidvarious Dynamics). shapes of vortex These shedder models arebar mainlywere determined. based on Reynolds theory. It is assumed that fluid motion can be defined by two parameters such as average velocity and fluctuation velocity.2. Numerical As a result, Method the Navier–Stokes equation describing the motion of a fluid additionally containsMany a turbulencemathematical stress models tensor are [evaluated12]. This causesto simulate additional flow velocity. variables The to most appear popu- in the equation.lar models On for correctness modeling complex of the numerical flow problems results are affects numerical using me suitablethods CFD turbulence (Computa- model andtional preparation Fluid Dynamics). right preprocessing These models are parameters. mainly based To someon Reynolds of themost theory. important It is assumed aspects includethat fluid the motion number can ofbe meshdefined elements by two parameters and the areas such of as its average density. velocity Conducted and fluctua- research purposetion velocity. the selection As a result, of the the Navier–Stokes optimal turbulence equation model describing and to the determine motion of thea fluid minimum ad- numberditionally of contains the mesh a elements.turbulence stress tensor [12]. This causes additional variables to ap- pearThe in the appearance equation. On of additionalcorrectness variablesof the numerical in the Reynoldsresults affects equations using suitable requires turbu- supple- mentinglence model the matamaticand preparation model right with preprocess additionaling equations. parameters. In practice,To some theyof the are most most im- often knownportant asaspects k-ε or include k-ω turbulence the number models of mesh [12 ,elements14]. Each and of thesethe areas models of its has density. certain Con- limita- tions.ducted They research make purpose each of the them selection reflect of the the real optimal movement turbulence of fluids model better and orto determine worse. In the firstthe minimum part of the number work, researchof the mesh was elements. carried out to identify the model that best reproduces the realThe shape appearance of the vonof additional Karman vortexvariables path. in the Reynolds equations requires supple- mentingThe the k-ε matamaticmodel takes model into with account additional the relationshipequations. In betweenpractice, they the vorticityare most often and the kineticknown energyas k-ε or of k- turbulenceω turbulence [14 ].models This model[12,14]. models Each of well these turbulence models has in certain free flow limita- and in sheartions. layers.They make It is each characterized of them reflect by low thesensitivity real movement to boundary of fluids better conditions, or worse. especially In the to thefirst distribution part of the work, of liquid research velocity was carried at the inletout to to identify the channel. the model This that is abest desirable reproduces feature the real shape of the von Karman vortex path. due to the fact that these quantities are often not exactly known in practical calculations. The k-ε model takes into account the relationship between the vorticity and the ki- Meanwhile, the k-ω model takes into account both the kinetic energy of turbulence and netic energy of turbulence [14]. This model models well turbulence in free flow and in the speed of energy dissipation in its solution. In the k-ω model, like in k-ε, k will express shear layers. It is characterized by low sensitivity to boundary conditions, especially to the kinetic energy of turbulence and ω is the ratio of the energy dissipation speed ε to the distribution of liquid velocity at the inlet to the channel. This is a desirable feature due the kinetic energy k. The k-ω model, on the other hand, models the turbulent flow in the to the fact that these quantities are often not exactly known in practical calculations. Mean- boundarywhile, the layerk-ω model much better,takes into while account it is very both sensitive the kinetic tothe energy values of ofturbulence turbulent and quantities the inspeed free flow.of energy dissipation in its solution. In the k-ω model, like in k-ε, k will express the kineticIt was energy found of turbulence that you can and combine ω is the the ratio desired of the features energy ofdissipation both models speed by ε combining to the themkinetic into energy a single k. The model, k-ω model, in ANSYS on the Fluent other it washand, named models k- theω SST. turbulent The k- flowω SST in modelthe wasboundary used for layer further much numerical better, while research. it is very This sensitive model to is the a good values representation of turbulent quantities of the kinetic energyin free flow. of the vortices forming. The appropriate selection of the finite element mesh has a significant impact on the simulation results [15]. The structural computational mesh was generated for rectangular

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It was found that you can combine the desired features of both models by combining It was found that you can combine the desired features of both models by combining them into a single model, in ANSYS Fluent it was named k-ω SST. The k-ω SST model them into a single model, in ANSYS Fluent it was named k-ω SST. The k-ω SST model was used for further numerical research. This model is a good representation of the kinetic Sensors 2021, 21, 4697 was used for further numerical research. This model is a good representation of the kinetic5 of 20 energy of the vortices forming. energy of the vortices forming. The appropriate selection of the finite element mesh has a significant impact on the The appropriate selection of the finite element mesh has a significant impact on the simulation results [15]. The structural computational mesh was generated for rectangular simulation results [15]. The structural computational mesh was generated for rectangular elementselements ofof variousvarious shapes and and sizes. sizes. The The mesh mesh is isrefined reﬁned in inthe the immediate immediate vicinity vicinity of the of elements of various shapes and sizes. The mesh is refined in the immediate vicinity of the thevortex vortex shedder shedder bar bar (Figure (Figure 4).4). For For the the purpos purposeses of of the research, aa calculationcalculation grid grid was was vortex shedder bar (Figure 4). For the purposes of the research, a calculation grid was preparedprepared for for a a rectangular rectangular duct duct with with a a length length of of L1 L1 = = 900 900 mm mm and and a a widthwidth of of L2L2 == 600600 mm.mm. prepared for a rectangular duct with a length of L1 = 900 mm and a width of L2 = 600 mm. AA cylindercylinder withwith aa diameter diameter of of 20 20 mm mm was was placed placed in in the the middle middle of of the the channel. channel. The The width width A cylinder with a diameter of 20 mm was placed in the middle of the channel. The width ofof thethe channelchannel isis soso largelarge thatthat thethe inﬂuenceinfluence ofofthe the side side walls walls of of the the channel channel on on the the process process of the channel is so large that the influence of the side walls of the channel on the process ofof vortexvortex formationformation cancan bebe neglected.neglected. of vortex formation can be neglected.

Figure 4. Computational mesh. FigureFigure 4. 4.Computational Computational mesh. mesh. Figure 5 presents the results of numerical simulation of the number of mesh of finite FigureFigure5 5presents presents the the results results of of numerical numerical simulation simulation of of the the number number of of mesh mesh of of ﬁnite finite 400,848 nodes. The k-ω SST model for a cylinder-shaped of shedder bar is a good repre- 400,848400,848 nodes.nodes. TheThe k-ω SST model for for a a cylinder-shaped cylinder-shaped of of shedder shedder bar bar is is a agood good repre- rep- sentation of the von Karman street, so it can be used in the study of shedder bars of other resentationsentation of of the the von von Karman Karman street, street, so soit can it can be used be used in the in thestudy study of shedder of shedder bars barsof other of shapes. othershapes. shapes.

Figure 5. The results of numerical calculations for the k-ω model used in the further part of the Figure 5. The results of numerical calculations for the k-ω model used in the further part of the Figurework. 5. The results of numerical calculations for the k-ω model used in the further part of the work. work. 3. Computational Results 3. Computational Results 3. ComputationalIn order to investigate Results the influence of the shape of the vortex generator on the stability In order to investigate the influence of the shape of the vortex generator on the sta- of theIn generated order to vortices,investigate simulations the influence were of carried the shape out for of variousthe vortex flow generator velocities on (0.5 the m/s, stability of the generated vortices, simulations were carried out for various flow velocities 1bility m/s, of 2 m/s).the generated Numerical vortices, calculations simulations were performed were carried with out the for ANSYS various Fluent flow software.velocities (0.5 m/s, 1 m/s, 2 m/s). Numerical calculations were performed with the ANSYS Fluent The(0.5 calculations m/s, 1 m/s, were2 m/s). carried Numerical out on calculations a computing were server performed consisting with of two the computers. ANSYS Fluent The software. The calculations were carried out on a computing server consisting of two com- firstsoftware. had 24 The cores calculations and the second were carried had 20 out cores on Ina computing numerical calculations,server consisting the Turbulenceof two computers. The first had 24 cores and the second had 20 cores In numerical calculations, the k-STputers. SST The model first was had used 24 cores for a and two-dimensional the second had velocity 20 cores field In andnumerical a non-stationary calculations, flow. the Turbulence k-ST SST model was used for a two-dimensional velocity field and a non-sta- TheTurbulence boundary k-ST conditions SST model were was the used assumption for a tw thato-dimensional there is a uniform velocity velocity field and distribution a non-stationary flow. The boundary conditions were the assumption that there is a uniform veloc- intionary the inflow flow. channel The boundary and that conditions the liquid were flowing the inassumption the channel that is there water. is a uniform velocity distribution in the inflow channel and that the liquid flowing in the channel is water. ity distributionThe calculation in the results inflow were channel compared and that with the the liquid results flowing for the in cylinder-shaped the channel is water. generator. For the digital genaric vortex, regular and stable vortices were obtained for a liquid flow velocity of 0.5 m/s, 1 m/s and 2 m/s. The results of the numerical calculations are presented in Table1. The frequency of the generated vortices was determined on the basis of how the pressure changed at the selected point on the basis of successive time steps.

The calculation results for other shapes of shedder bars will be compared to the values presented in Table1. Sensors 2021, 21, 4697 6 of 20

Table 1. Computational results for the vortex shedder from the Figure2.

v [m/s] f [Hz] ∆p [Pa] 0.5 6.01 128.11 1 12.44 442.51 2 26.32 1529.5

The main objective of the numerical research was to determine the effect of individual surfaces of the shedder bar on the frequency, pressure and stability of vortex formation. These are the parameters that have a direct impact on the proper operation of the vortex ﬂowmeter. The range of changes in the frequency of the generated vortices affects the sensitivity of the ﬂowmeter. Figure6 shows the numerical results of comparing the values of frequency changes of the vortices formed for various modifications of the vortex generator. The designations of the Sensors 2021, 21, x FOR PEER REVIEW 7 of 22 shapes correspond to the markings in Figure2. The values in the diagram represent the ratio of the frequency of vortices generated by individual shapes to the reference frequency f0 (Table1).

FigureFigure 6. Shapes6. Shapes resulting resulting from from (a ()a) rake rake surface surface modiﬁcationmodification (b) runoff surface surface modification modiﬁcation (c ()c oval.) oval.

The more the f/f0 ratio differs from 1, the greater the impact on the sensitivity of the flowmeter will be the modification of the surface areas consistent with the markings in Figure 6. As it results from Figure 6a, the greatest impact on the sensitivity of the vortex flowmeter is the modification of areas 1 and 3 together with areas 2 and 4. In this type, the greatest variations are obtained for different flow velocity, but the variations are non-linear. Modifying only areas 3 and 4 or 1 and 2 does not bring such good results. The modifications to the P3 and P4 deserve special attention here In the case of the runoff surface modification, a significant improvement in the flowmeter sensitivity is obtained for the shapes T2 and T7. These shapes have a sharp edge, which makes it easier to detach the vortices. Modification of the upper and lower flow surfaces of the vortex shedder bar (shapes in Figure 6c) mainly increases the frequency of the generated vortices, but does not significantly affect the sensitivity of the flowmeter. Only the modification of W3 has a visible effect on the sensitivity.

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The more the f/f0 ratio differs from 1, the greater the impact on the sensitivity of the flowmeter will be the modification of the surface areas consistent with the markings in Figure6. As it results from Figure6a, the greatest impact on the sensitivity of the vortex flowmeter is the modification of areas 1 and 3 together with areas 2 and 4. In this type, the greatest variations are obtained for different flow velocity, but the variations are non-linear. Modifying only areas 3 and 4 or 1 and 2 does not bring such good results. The modifications to the P3 and P4 deserve special attention here In the case of the runoff surface modification, a significant improvement in the flowmeter sensitivity is obtained for the shapes T2 and T7. These shapes have a sharp edge, which makes it easier to detach the vortices. Modification of the upper and lower flow surfaces of the vortex shedder bar (shapes in Figure6c) mainly increases the frequency of the generated vortices, but does not significantly affect the sensitivity of the flowmeter. Only the modification of W3 has a visible effect on the sensitivity. The second important parameter influencing the operation of the vortex flowmeter is the value of pressure changes of the generated vortices. The greater the pressure change of the generated vortices, the greater the signal-to-noise ratio in the measurement signal. Figure7 shows the results of the numerical simulation. These results show the impact of modification of the rake and trailing surface on pressure changes generated by vortices formed behind individual shedder bars. The greatest changes for different fluid streams are obtained for modifications P3 and P4 (Figure7a) and for modifications T1 and T2 (Figure7b). The least effect is obtained for the T7 modification. On the other hand, for the W1 modification, the changes increase with small flow rate, while for larger flows they remain at the same level. Modification of W3 lowers the level by approximately a constant amount. Based on the research, it was found that the modification of the rake surface, consisting in increasing the drag coefficient, causes an increase in the pressure difference behind the vortex generator. This improves the signal-to-noise properties of the flowmeter, but the processing characteristics become more non-linear. This will make a difference at higher fluid flow rates. The stability of the von Karman street has a great influence on the correct work of the vortex flowmeter. Further research was carried out to determine the impact of changes in the shape of rake surface on the stability of vortex street. Modifications of the shape of shedder bar P3, P4, T1, T2, T7, W1 and W3 were selected for further numerical tests. The results of the analysis for the shapes P3 and P4 are shown in Figure8. Modification of the rake surface destabilizes the vortex street in a wide range of velocity of the tested fluid. The destabilization of the vortex street is already noticeable for a very low velocities of flow velocity v = 0.5 m/s (Figure8a,b). Even a gentle flattening of the rake face of the vortex sedder bar causes considerable disturbance to the formation of the vortex street. Modification P4 may be appropriate for very low flow rates as it generates the greatest amplification of the pressure amplitude downstream of the vortex generator. Moreover, it extends the measuring range. The analysis shows that in order to obtain good stability of the generated vortices, it is best if the rake surface is as streamlined as possible. It will be a good solution especially for measuring high velocities of flowing fluid. Analyzing the obtained results for the shapes in which the runoff surfaces T1 and T2 were modified (Figure9a,b), it was found that the modification of these surfaces did not destabilize the stream. The reason is that the stream of flowing liquid mainly breaks off from the top and bottom surfaces of the shedder bar. These modifications advantageously affect the amplitude of pressure changes of the resulting vortices. The T7 modification (Figure9e,f) has an even smaller impact on the stability of the von Karman swirl path. Since Modification T7 has little effect on improving the value of the pressure amplitude of the resulting vortices, it is advisable to pay special attention only to the modification T2. Sensors 2021, 21, 4697 8 of 20 Sensors 2021, 21, x FOR PEER REVIEW 8 of 22

The streamlined shape of the rake surface has a positive effect on the stabilization Theof the second vortex streetimportant for high parameter flow velocities. influencing Sharp ending the of operation the trailing of surface the vortex does not flowmeter is thedestabilize value of pressure the vortex changes street, but of improves the generated the sensitivity vortices. of the The vortex greater flowmeter. the pressure Thus, it change has a positive effect on improving the parameters of the vortex flowmeter. To increase. The of themodification generated of vortices, W3 does notthe significantlygreater the change signal-to-noise the parameters ratio of thein the vortex measurement flowmeter; signal. Figurehowever, 7 shows it linearizes the results its characteristics. of the numerical Based onsimulation. this analysis, These for experimental results show research, the impact of modificationit was decided of the to rake develop and the trailing shapes surface of vortex on generators pressure which changes constitute generated a kind ofby vortices formedcompilation behind individual of the modifications shedder presented bars. in Figure2. Figure 10 shows the proposed shapes of vortex shedder bars that have been subjected to experimental tests.

Figure 7.Figure Shapes 7. Shapes resulting resulting from from (a) (rakea) rake surface surface modification modiﬁcation (b )(b runoff) runoff surface surface modiﬁcation modification (c) oval. (c) oval.

The greatest changes for different fluid streams are obtained for modifications P3 and P4 (Figure 7a) and for modifications T1 and T2 (Figure 7b). The least effect is obtained for the T7 modification. On the other hand, for the W1 modification, the changes increase with small flow rate, while for larger flows they remain at the same level. Modification of W3 lowers the level by approximately a constant amount. Based on the research, it was found that the modification of the rake surface, consisting in increasing the drag coefficient, causes an increase in the pressure difference behind the vortex generator. This improves the signal-to-noise properties of the flowmeter, but the processing characteristics become more non-linear. This will make a difference at higher fluid flow rates.

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FigureSensors 8. TheFigure 2021 inﬂuence, 21 8., x The FOR influence PEER of 1, REVIEW 2, of 3 and1, 2, 3 4 and surfaces 4 surfaces modiﬁcation modification onon regularity of ofgenerated generated vortices vortices for (a) P3 for with (a) v P3 = 0.5 with m/s,11 v =of 0.522 m/s, (b) P3 with v = 2 m/s, (c) P4 with v = 0.5 m/s, (d) P4 with v = 2 m/s. (b) P3 with v = 2 m/s, (c) P4 with v = 0.5 m/s, (d) P4 with v = 2 m/s.

Figure 9. TheFigure inﬂuence 9. The influence of 5,6,7 of and 5,6,78 and surfaces 8 surfaces modiﬁcation modification onon regularity of ofgenerated generated vortices vortices for (a) forT1 with (a) T1v = with0.5 m/s, v = 0.5 m/s, (b) T1 with(b v) T1 = 2with m/s, v = (2c )m/s, T2 ( withc) T2 with v = 0.5v = 0.5 m/s, m/s, (d (d)) T2 T2 withwith v v = =2 m/s, 2 m/s, (e) T2 (e )with T2 v with = 0.5v m/s, = 0.5 (f) T2 m/s, with (f v) = T2 2 m/s. with v = 2 m/s.

Figure 10. Experimental shapes of vortex shedder bars.

4. Test Stand for Experimental Research In order to verify the numerical calculations and determine the influence of the dimensions and shapes of the vortex shedder bar on the characteristic dimensions related to the construction of vortex flow meters, a test stand was built and a measurement method was developed to enable the visualization of vortex’s arising behind the shedder. In order to visualize the vortex path, the marker method was used [16]. This method enables the measurement of the local velocity of fluid movement and the tracing of eddies that occur. The selection of tags has a great influence on the results of the visualization.

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Figure 9. The influence of 5,6,7 and 8 surfaces modification on regularity of generated vortices for (a) T1 with v = 0.5 m/s, (b) T1 with v = 2 m/s, (c) T2 with v = 0.5 m/s, (d) T2 with v = 2 m/s, (e) T2 with v = 0.5 m/s, (f) T2 with v = 2 m/s.

FigureFigure 10. ExperimentalExperimental shapes shapes of of vortex shedder bars. 4. Test Stand for Experimental Research 4. Test Stand for Experimental Research In order to verify the numerical calculations and determine the influence of the In order to verify the numerical calculations and determine the influence of the di- dimensions and shapes of the vortex shedder bar on the characteristic dimensions related mensions and shapes of the vortex shedder bar on the characteristic dimensions related to the construction of vortex flow meters, a test stand was built and a measurement method to the construction of vortex flow meters, a test stand was built and a measurement was developed to enable the visualization of vortex’s arising behind the shedder. Sensors 2021, 21, x FOR PEER REVIEWmethod was developed to enable the visualization of vortex’s arising behind the shedder.12 of 22 In order to visualize the vortex path, the marker method was used [16]. This method In order to visualize the vortex path, the marker method was used [16]. This method enables the measurement of the local velocity of fluid movement and the tracing of eddies enables the measurement of the local velocity of fluid movement and the tracing of eddies that occur. The selection of tags has a great influence on the results of the visualization. thatThey occur. should The be selection of an appropriate of tags has size a greatand the influence material on from the whichresults they of the are visualization. made should They should be of an appropriate size and the material from which they are made should have a density similar to that of the fluid. These markers should not significantly interfere have a density similar to that of the fluid. These markers should not significantly interfere with fluid movement. Water with polyamide markers with a density of 940 kg/m3 and with fluid movement. Water with polyamide markers with a density of 940 kg/m3 and granulation from 20 µμmm toto 11 mmmm waswas usedused forfor thethe experimentalexperimental tests.tests. TheThe markersmarkers hadhad aa good reflectivereflective surface.surface. The measurement method presented in this article is similar to the PIV. However,However, itit differs inin thethe methodmethod ofof calculating calculating the the local local speed speed value. value. This This method method consists consists in in register- regis- ingtering the the trajectory trajectory of theof the movement movement of markers of markers moving moving with with the liquid.the liquid. Compared Compared to the to PIVthe PIV method, method, the tagsthe tags have have much much larger larger dimensions. dimensions. This kindThis low-costkind low-cost DPIV set-ups,DPIV set-ups, using freeusing PIV free software PIV software and high-speed and high-speed cameras cameras instead ofinstead the classical of the classical double exposure double exposure cameras, havecameras, been have successfully been successfully employed inemployed PIV and liquid-granularin PIV and liquid-granular PIV applications PIV [applications17–19]. The basis[17–19]. for The calculating basis for the calculating speed of movement the speed of of the movement tag is the analysisof the tag of is its the movement analysis based of its onmovement the image based of only on the one image image of ofonly the one fluid imag flow.e of This the fluid method flow. does This not method require does image not correlationrequire image of atcorrelation least two of consecutive at least two photos, consecutive as is the photos, case withas is thethe PIVcase method.with the ThePIV imagemethod. of The a moving image particle of a moving is recorded particle with is recorded a long exposure. with a long The exposureexposure. time The isexposure so long thattime theis so marker long that in the the imagemarker is in visible the image in the is form visible of in a longitudinal the form of a streak longitudinal with a shapestreak correspondingwith a shape corresponding to the marker to movement the marker trajectory. movement trajectory. The local liquid velocity is determineddetermined on the basisbasis ofof thethe tracestraces leftleft byby thethe markersmarkers (Figure(Figure 1111).). AfterAfter hittinghitting the the marker, marker, the the light light that that shines shines through through the the measurement measurement section sec- istion reflected is reflected from from its surface. its surface. In cameos In cameos it is visible it is visible as a bright as a bright spot. Whenspot. When a long a exposure long ex- timeposure is used,time is the used, trace the of trace the moving of the moving marker marker will be visiblewill be as visible a bright as a line. bright line.

FigureFigure 11.11. ImageImage ofof tracertracer particleparticle ﬂow.flow.

The length of the trace left by the marker in the image is the basis for calculating the local velocity and direction of movement of the fluid. The local speed is calculated from the formula:

⋅ 𝑣= , (3) where: v—velocity of liquid flow, k—scale coefficient, p—number of pixels, t—exposition time. On the basis of Equation (3), the instantaneous values of the fluid movement are calculated. This method determines the instantaneous velocities at various points in the flow and at various distances from the vortex generator. In the further part of the work, the velocities of the liquids refer to the values averaged from the instantaneous values in the entire observed cross-section. The longer the exposure time and the higher the speed of liquid movement, the longer the streak recorded by the moving marker will be (Figure 12). Long marks are advantageous for the accuracy of the measurement. However, due to the fact that the markers move in three planes for long tracks, it often happens that the marker leaves the space

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The length of the trace left by the marker in the image is the basis for calculating the local velocity and direction of movement of the fluid. The local speed is calculated from the formula: k·p v = , (3) t where: v—velocity of liquid flow, k—scale coefficient, p—number of pixels, t—exposition time. On the basis of Equation (3), the instantaneous values of the fluid movement are calculated. This method determines the instantaneous velocities at various points in the flow and at various distances from the vortex generator. In the further part of the work, the velocities of the liquids refer to the values averaged from the instantaneous values in the entire observed cross-section. The longer the exposure time and the higher the speed of liquid movement, the Sensors 2021, 21, x FOR PEER REVIEW 13 of 22 longer the streak recorded by the moving marker will be (Figure 12). Long marks are advantageous for the accuracy of the measurement. However, due to the fact that the markers move in three planes for long tracks, it often happens that the marker leaves the illuminatedspace illuminated by the by laser. the laser.Therefore, Therefore, there thereis an isoptimal an optimal exposure exposure time time for the for theselected selected in- tervalinterval of offluid fluid movement movement velocity. velocity.

Figure 12. SampleSample images of registered trace markersmarkers forfor ((aa)) exposureexposure timetime 1/10 1/10 s of particleparticle velocityvelocity 0.7990.799 m/s,m/s, ( (b)) exposure exposure time 1/50 s s of of particle particle velocity velocity 0.047 0.047 m/s. m/s.

ForFor the the purpose purpose of of the the research, the optimal exposure times for the velocity of liquid movementmovement were determined in the measuremen measurementt channel. The influence influence of exposure time onon the uncertainty of of measurement of of the velo velocitycity of of liquid movement was was investigated. investigated. The uncertainty increases both when the exposure time is too long and also when it is too short.short. In In the the case case of of short short exposure exposure times, times, th thee measurement measurement error is mainly affected by the resolutionresolution of the recorded images. This is due due to the formation of very short trace that are leftleft by by mud flakes. flakes. Since Since the the camera camera has has limited limited possibilities possibilities of of changing changing the the resolution, resolution, thethe measurement measurement uncertainty uncertainty is is optimized optimized by by me meansans of of a a properly properly selected selected exposure exposure time. time. BasedBased on on experimental experimental studies, studies, the the recommended recommended exposure exposure times times for for selected selected veloc- veloc- itiesities of of fluid fluid movement movement were were determined. determined. A number A number of measurements of measurements were were made made for dif- for ferentdifferent exposure exposure times times for for markers markers moving moving in in the the speed speed range range from from 0,01 m/s to to 0,65 m/s. The tests tests were were carried carried out out for for the the following following exposure exposure times times 1/5 1/5 s, s,1/10 1/10 s, 1/20 s, 1/20 s, 1/30 s, 1/30 s and s 1/50and 1/50s. The s. uncertainty The uncertainty for a Type for a TypeA was A calculated was calculated based based on the on formula the formula [20]: [20]:

s N∑()2 𝑈=s =∑n=1 (vi − vs) , (4) U = √ =√ () , (4) N N(N − 1) where: s—estimated standard deviation, vi—flow velocity, vs—mean flow velocity, N— numberwhere: s of—estimated measurements. standard deviation, vi—ﬂow velocity, vs—mean ﬂow velocity, N— numberThe ofmean measurements. flow velocity is calculated on equation [20]: 1 𝑣 = 𝑣 (5) 𝑁 Figure 13 shows the results of research on the influence of exposure time on the uncertainty of the measurement of the speed of movement. To make the presented results easier to read, the uncertainty values were expressed as a percentage. The percentages represent the ratio of uncertainty to the average velocity of fluid movement. Calculated based on the formula 𝑈[%] = ⋅ 100%, (6)

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The mean ﬂow velocity is calculated on equation [20]:

1 N vs = ∑ vi (5) N n=1

Figure 13 shows the results of research on the inﬂuence of exposure time on the uncertainty of the measurement of the speed of movement. To make the presented results easier to read, the uncertainty values were expressed as a percentage. The percentages represent the ratio of uncertainty to the average velocity of ﬂuid movement. Calculated based on the formula U Sensors 2021, 21, x FOR PEER REVIEW = · 14 of 22 U[%] 100%, (6) vs

35 1/5 1/10 30 1/20 25 1/30 1/50 20

0 10 40 160 640 FigureFigure 13.13.The The resultsresults ofof thethe calculationcalculation ofof thethe measurementmeasurement uncertaintyuncertainty forfor differentdifferent exposure exposure times times andand the the speed speed of of the the marker marker movement. movement.

AsAs itit resultsresults from from the the measurements, measurements, exposure exposure times times below below 1/10 1/10 ss areare recommendedrecommended toto bebe used used for for liquid liquid movement movement speeds speeds below below 100 100 mm/s. mm/s. However,However, for for liquid liquid movement movement speedsspeeds belowbelow 500500 mm/s, mm/s, it is recommended toto useuse exposureexposure timestimes shortershorter thanthan 1/301/30 s. TheThe accuracyaccuracy of of thethe speedspeed measurementmeasurement isis significantlysignificantly influenced influenced by by the the appropriateappropriate selectionselection ofof thethe exposureexposure time. time. InIn orderorder toto selectselect the the optimal optimal exposure exposure time, time, studies studies were were carriedcarried outout onon thethe influenceinfluence ofof exposureexposure timetime on on the the error error of of local local velocity velocity measurement measurement dependingdepending onon the liquid velocity [21]. [21]. Based Based on on experimental experimental studies, studies, exposure exposure times times for fordifferent different velocities velocities of fluid of fluid movement movement have have been been proposed proposed (Table (Table 2).2 ).By By analyzing analyzing the theresults results in Table in Table 2, it2 ,can it can be concluded be concluded that that there there are at are least at least two tworecommended recommended exposure exposuretimes for times most for measuring most measuring ranges. ranges.While there While is some there flexibility is some flexibility in the choice in the of choice exposure of exposuretimes, there times, are theresome arethreshold some threshold values that values should that not should be exceeded not be exceeded due to the due measure- to the measurementment uncertainty. uncertainty.

TableTable 2. 2.Recommended Recommended exposition exposition times times depending depending on on the the liquid liquid velocities velocities [ 21[21].].

RecommendedRecommended Exposition Exposition Time Time [s] [s] Mean Mean Velocity Velocity of Tracing of Tracing Particles Particles [m/s] [m/s] 1/51/5 <0.1 <0.1 1/101/10 <0.15 <0.15 1/201/20 >0.1 >0.1 1/30 >0.15 1/30 >0.15 1/50 >0.4 1/50 >0.4

As it results from the conducted analysis, the inaccuracies in measuring the velocity of the liquid when selecting the appropriate exposure time are in the order of a few percent. These speeds were compared with those calculated on the basis of the measurement of the liquid stream flowing in the channel. The liquid flow was measured with a standardized measuring orifice. When comparing the results, a similar accuracy was obtained and the discrepancy of the results did not exceed 5%. Such accuracy was found to be sufficient for the purposes of this study. From the consecutive photos it is possible to follow the motion of the vortex path (Figure 14). By analyzing consecutive photos, it is possible to determine when the maximum turbulence occurs at a selected distance from the vortex shedder. For example, in Figure 14a there is an observation point in which the maximum of the occurring vortex is visible for 66 frames of the image. Then, for the frame 76, this maximum is formed on the other side of the horizontal axis (Figure 14b). Again, the vortex maximums for the selected

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As it results from the conducted analysis, the inaccuracies in measuring the velocity of the liquid when selecting the appropriate exposure time are in the order of a few percent. These speeds were compared with those calculated on the basis of the measurement of the liquid stream flowing in the channel. The liquid flow was measured with a standardized measuring orifice. When comparing the results, a similar accuracy was obtained and the discrepancy of the results did not exceed 5%. Such accuracy was found to be sufficient for the purposes of this study. From the consecutive photos it is possible to follow the motion of the vortex path (Figure 14). By analyzing consecutive photos, it is possible to determine when the maximum Sensors 2021, 21, x FOR PEER REVIEW 15 of 22 turbulence occurs at a selected distance from the vortex shedder. For example, in Figure 14a there is an observation point in which the maximum of the occurring vortex is visible for 66 frames of the image. Then, for the frame 76, this maximum is formed on the other side ofviewpoint the horizontal will appear axis (Figure in the 1486b). frame Again, of the the im vortexage (Figure maximums 14c). forOn the this selected basis, the viewpoint number willof frames appear between in the 86 successive frame of thevortex image maxima (Figure is 14calculated.c). On this Since basis, the the camera number captures of frames the betweenimage at successive a constant vortex frame maxima rate, the is duration calculated. of Sincethe vortices the camera can capturesbe calculated the image fromat the a constantframe rate. frame The rate, frequency the duration of the ofresulting the vortices vortices can bewas calculated calculated from on thethe framebasis of rate. the The pe- frequencyriod. of the resulting vortices was calculated on the basis of the period. 1 1 N 𝑁 f = = F (7) 𝑓 T= =(t − t ) (7) 𝑇 2(𝑡 2−𝑡)

where:where: t2,, t t00—number—number of of the the image image frame frame in inwhich which the the vortex vortex maxima maxima appear appear at the at theob- observedserved point, point, NFN—numberF—number of offrames frames of ofthe the recorded recorded image image per per second. second.

0 i 2 FigureFigure 14. 14.Sequence Sequence of of consecutive consecutive photos photos (a ()a t)0 t== 66, 66, ( b(b)) t it=76,=76, ((cc)) tt2 = 86. 86.

InIn order to to conduct conduct experimental experimental research, research, a measuring a measuring stand stand was was designed designed and built and built(Figure (Figure 15). The 15). resulting The resulting vortices vortices are observed are observed in a transparent in a transparent channel channel (1) with (1) a withrectan- a rectangulargular cross-section cross-section and dimensions and dimensions of 50.0 of 50.0× 250× 250mm. mm. The Thechannel channel is entirely is entirely made made of ofacrylic acrylic glass. glass. In its In upper its upper part, part, a cover a cover is mounted, is mounted, thanks thanks to which to which it is possible it is possible to mount to mountvarious various vortex generators vortex generators in the measuring in the measuring space. The space. water The in water the channel in the channelis in a closed is in acircuit closed and circuit is stored and isin storeda tank in(2) a with tank a (2) capacity with a of capacity 1 m3. The of 1liquid m3. Theis pumped liquid is by pumped a pump 3 by(3) awith pump a capacity (3) with of a capacity17 m3/h and of 17 a maximum m /h and ahead maximum of 18 m. head The ofpump 18 m. is Thedriven pump by a is 3 drivenkW electric by a 3motor. kW electric The water motor. flow The in waterthe cha flownnel in is the regulated channel by is the regulated throttle by valve the throttle(5). The valvevalve (5).(4) is The used valve to regulate (4) is used the to pressure regulate in the the pressure duct. In inorder the duct.to obtain In order the same to obtain local thevelocities same local at the velocities inlet to atthe the measuring inlet to the channel, measuring a flow channel, straightener a flow straightener (6) was used. (6) wasThe used.straightener The straightener is shaped to is condition shaped tothe condition flow by increasing the flow bythe increasingflow velocity the along flow the velocity chan- alongnel walls the and channel reducing walls turbulence. and reducing In the turbulence. middle of In its the length, middle there of itsis length,an additional there isbun- an additionaldle-pipe straightener. bundle-pipe Above straightener. the measuring Above thesection, measuring the Sony section, ILCE-7SM2 the Sony digital ILCE-7SM2 camera was installed, which recorded the movement of the markers. The camera allows you to record movies with Full HD resolution and a maximum speed of 120 fps. The camera allows you to set the exposure time in the range from 1/8000 s to 30 s.

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Sensors 2021, 21, x FOR PEER REVIEW 16 of 22 digital camera was installed, which recorded the movement of the markers. The camera Sensors 2021, 21, x FOR PEER REVIEW 16 of 22 allows you to record movies with Full HD resolution and a maximum speed of 120 fps. The camera allows you to set the exposure time in the range from 1/8000 s to 30 s.

Figure 15. Experimental setup. Figure 15. Experimental setup. The experimental tests were conducted over a length of 450 mm. The measuring sectionTheThe was experimental illuminated tests in awere horizontal conducted plane ov over fromer a two length directions ofof 450450 mm. mm.with Thethe measuringuse of line sectionlasers.(Figuresection was was illuminated illuminated 16) The lasers in in a ahad horizontal a peak poplane planewer from fromof 1 Wtwo and directions the emitted with wavelength the use of was line lasers.(Figure520lasers.(Figure nm. They 16 16)were) TheThe semiconductor laserslasers had had a a peak peak lasers power po equippedwer of of 1 W1 W andwith and the a thecollimator emitted emitted wavelength generatingwavelength was a wasline 520 520withnm. nm. Theya scattering They were were semiconductor lens. semiconductor The use of lasers lighting lasers equipped from equipped both with sides awith collimator significantly a collimator generating improved generating a line the witha pro-line a withjectionscattering a scatteringof the lens. particles, The lens. use The as of it use lighting reduced of lighting from the appearance bothfrom sides both signiﬁcantlysides of a darksignificantly shadow improved improvedbehind the the projection the illumi- pro- jectionnatedof the particle. particles,of the particles, The as lasers it reduced as wereit reduced theinstalled appearance the alongappearance the of a sides dark of a of shadowdark the shadowchannel behind inbehind thesuch illuminated thea way illumi- that natedtheparticle. liquid particle. The film lasers The with lasers were a thickness were installed installed ofalong 1mm along thewas sides theilluminated sides of the of channelthe in channelthe inmiddle such in such aof way the a way thatchannel that the theheight.liquid liquid ﬁlm film with with a thickness a thickness of 1mm of 1mm was illuminated was illuminated in the middlein the middle of the channel of the channel height. height.

Figure 16. Experimental section. Figure 16. Experimental section. 5. Validation of the Results of Numerical Calculations withwith thethe ExperimentExperiment ResultsResults 5. ValidationA series of experimentalthe Results of studies Numerical were Calculations carried out to with investigate the Experiment the influenceinfluence Results of the shape of the vortex shedder bar of the assumptions made for the numerical calculations, shapeA ofseries the vortexof experimental shedder bar studies of the were assu carriedmptions out made to investigate for the numerical the influence calculations, of the the resultsresults ofof thethe numerical numerical calculations calculations were were compared compared with with the the results results of the of experiment.the experi- shapeThe basic of the value vortex measured shedder in vortexbar of flowthe assu metersmptions is the made vortex for frequency; the numerical therefore, calculations, the basic thement. results The basicof the value numerical measured calculations in vortex were flow comparedmeters is the with vortex the resultsfrequency; of the therefore, experi- thevalidation basic validation criterion was criterion the change was the of the change frequency of the of frequency the generated of the vortices generated from thevortices flow ment.velocity. The Figure basic 17value shows measured the results in vortex for a cylinder-shapedflow meters is the shedder. vortex frequency; therefore, thefrom basic the flowvalidation velocity. criterion Figure was 17 shows the change the results of the for frequency a cylinder-shaped of the generated shedder. vortices from Figurethe flow 17 velocity.shows the Figure dependence 17 shows of thethe resultsfrequency for aof cylinder-shaped the generated vortices shedder. as a function ofFigure the velocity 17 shows of the fluid. dependence Particular of re thelationships frequency were of the determined generated for vortices various as diame-a func- tionters of thecylinder. velocity The of red fluid. dotted Particular line represen relationshipsts the wereresults determined obtained from for various the numerical diame- terssimulation of cylinder. for a cylinderThe red withdotted a diameterline represen of 20ts mm. the results obtained from the numerical simulation for a cylinder with a diameter of 20 mm.

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FigureFigure 17. 17. TheThe dependence dependence of of the the frequency frequency of of the the generate generatedd vortices vortices on on the the velocity velocity of of fluid ﬂuid movement. movement.

TheFigure continuous 17 shows lines the approximated dependence of the the re frequencysults of experimental of the generated measurements vortices asfor a variousfunction dimensions of the velocity of the of vortex fluid. generator. Particular By relationships comparing the were obtained determined results, for a satisfac- various torydiameters convergence of cylinder. of the The results red dotted was lineobtained represents. The the results results of obtained the numerical from the simulation numerical showsimulation a tendency for a cylinderto overestimate with a diameter the frequenc of 20y mm.of the generated vortices. The reason for this mayThe be continuous the approximations lines approximated resulting the from results the ofturbulence experimental model measurements used. This inaccu- for var- racyious can dimensions be corrected of the by vortex introducing generator. an Byappropriate comparing value the obtained of the constant results, acomponent. satisfactory Anotherconvergence important of the parameter results was is obtained.the angle Theof the results solid ofline. the This numerical angle is simulation the greater show the smallera tendency the size to overestimate of the generator. the frequencyThe greater of the the angle generated of inclination, vortices. the The greater reason the for sen- this sitivitymay be of the the approximations flowmeter. However, resulting the fromuse of the small turbulence dimensions model of the used. generator This inaccuracy limits the can be corrected by introducing an appropriate value of the constant component. Another range of applicability as it is required that the Reynolds number should not be less than important parameter is the angle of the solid line. This angle is the greater the smaller 5000 for the entire measuring range. Analyzing the slope of the dashed theoretical line, it the size of the generator. The greater the angle of inclination, the greater the sensitivity is greater than it would appear from the experimental measurements. The reason is that of the flowmeter. However, the use of small dimensions of the generator limits the range the numerical tests do not consider the influence of the side walls. of applicability as it is required that the Reynolds number should not be less than 5000 Since the numerical tests described in this article were aimed at determining how for the entire measuring range. Analyzing the slope of the dashed theoretical line, it is individual surfaces of the vortex shedder bar affect the parameters of the flowmeter and greater than it would appear from the experimental measurements. The reason is that the the results were related to the cylinder-shaped generator, the obtained convergence of the numerical tests do not consider the influence of the side walls. numerical results with the experiment results can be considered sufficient. The basic cri- Since the numerical tests described in this article were aimed at determining how terion for the selection of a vortex generator is the linearity of the frequency velocity de- individual surfaces of the vortex shedder bar affect the parameters of the flowmeter and pendence. As can be seen from Figure 1, the frequency of the resulting vortices depends the results were related to the cylinder-shaped generator, the obtained convergence of linearlythe numerical on the velocity results with of the the floating experiment liquid. results This tendency can be considered is consistent sufficient. with the The theoret- basic icalcriterion relationship for the expressed selection ofby athe vortex formula generator (1). On isthis the basis, linearity the corr of theectness frequency of the assump- velocity tionsdependence. resulting As from can the be seentheoretical from Figure analysis1, the is frequencyconfirmed, of which the resulting assumptions vortices were depends used tolinearly develop on specific the velocity shapes of the of vortex floating generato liquid. Thisrs, which tendency were is further consistent tested with experimentally. the theoretical relationshipThe range expressed of frequency by the changes formula (1).strongly On this de basis,pends the on correctnessthe characteristic of the assumptions size of the generator.resulting fromThe characteristic the theoretical dimension analysis also is confirmed, affects the whichslope of assumptions the characteristic were and used the to valuedevelop of the specific zero shapespoint. The of vortex zero factor generators, k is in which practice were only further determined tested experimentally. during the first calibrationThe range and does of frequency not change changes throughout strongly the service depends life. on The the value characteristic of the flow size velocity of the isgenerator. calculated The from characteristic the relationship dimension where also the affectsratio d/St the is slope the ofslope the of characteristic the linear depend- and the encevalue v of= f the(f): zero point. The zero factor k is in practice only determined during the first calibration and does not change throughout the service life. The value of the flow velocity

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is calculated from the relationship where the ratio d/St is the slope of the linear dependence 𝑑 v = f (f): 𝑣=𝑓 +𝑘 (8) d 𝑆𝑡 v = f + k (8) where: St—Strouhal number, f—frequency ofSt generated vortices, d—characteristic dimen- where:sion ofSt the—Strouhal obstacle, number,k—zero factorf —frequency of generated vortices, d—characteristic dimension ofTable the obstacle, 3 presentsk—zero functions factor approximating the results of the experimental data pre- sentedTable in 3Figure presents 17. functionsBased on these approximating functions, theit is results possible of to the calculate experimental the values data of pre- the sentedStrouhal in Figurenumber 17 for. Based a shedder on these bar an functions,d the value it is of possible the k coefficient. to calculate the values of the Strouhal number for a shedder bar and the value of the k coefﬁcient. Table 3. Mathematical functions approximating the results of the experimental. Table 3. Mathematical functions approximating the results of the experimental. d [mm] Mathematical Function of Frequency 12d [mm] Mathematicalf = 0.0152 Function∗ v + 1.7054 of Frequency 18 12 f = 0.0109f = 0.0152 ∗ v∗ +v 0.5273 + 1.7054 25 18 f = 0.0086f = 0.0109 ∗ v∗ +v 0.0769 + 0.5273 ∗ 25 f = 0.0086∗ v + 0.0769 30 30 f = 0.0073f = 0.0073 v∗ −v 0.0333− 0.0333 38 38 f = 0.0061f = 0.0061 ∗ v∗ −v 0.2214− 0.2214

Figure 18 shows the dependence of the value of the Strouhal number on the value of Figure 18 shows the dependence of the value of the Strouhal number on the value of the characteristic dimension. The linearity of this relationship proves that the measure- the characteristic dimension. The linearity of this relationship proves that the measurements were correctly carried out. Similar compliance with the theoretical assumptions is ments were correctly carried out. Similar compliance with the theoretical assumptions is demonstrated by the linear nature of the d/St slope coefficient. As shown by the performed demonstrated by the linear nature of the d/St slope coefficient. As shown by the performed measurements, the coefficient of zero k is a non-linear dependence on the characteristic measurements, the coefficient of zero k is a non-linear dependence on the characteristic dimension.dimension. Hence,Hence, itit isis necessarynecessary toto determinedetermine thisthis coefficientcoefficient individuallyindividually forfor eacheach type,type, shapeshape oror sizesize change of of the the vortex vortex shedder shedder bar. bar. For For the the generator generator used used in the in experiment, the experiment,the translation the translation function function of calculating of calculating the k the-factork-factor depending depending on the on diameter the diameter of the of gen- the generatorerator is: is: k𝑘= 0.18 = 0.18 log(d −log(𝑑−6.56.5) − 0.24) −0.24 (9)(9)

FigureFigure 18. 18.Inﬂuence Influence of of the the characteristic characteristic dimension dimension on on the the parameters parameters of of Equation Equation (3). (3).

BasedBased onon the the analysis analysis of of the the image image of of the the von von Karman Karman vortex vortex street, street, it it is is possible possible to to determinedetermine two two basic basic characteristic characteristic dimensions dimensions (Figure (Figure 19). 19). The The most most important important dimensions dimen- aresions the are length the length of the vortexof the vortex zone l zoneand thel and deviation the deviation from thefrom axis theh .axis These h. These dimensions dimen- dependsions depend on the sizeon the and size shape and of shape the shedder of the shedder bar. These bar. dimensions These dimensions are important are important as they

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Sensors 2021, 21, 4697 17 of 20 as they determine the optimal location for install a vortex detector. In addition, the dimen- sionalas they ratio determine h/l is a the criterion optimal for location the stability for inst ofall vortices. a vortex It detector. is believed In addition, that it should the dimen- be in thedeterminesional range ratio from the h/l optimal is0.15 a criterion to location0.45 [22].for forthe Figure installstability 19 a showsvortex of vortices. an detector. exemplary It is Inbelieved addition, image that of the ait von dimensionalshould Karman be in streetratiothe range hcaptured/l is from a criterion with0.15 toa camera. for0.45 the [22]. stabilityThe Figure image of 19 vortices.sh showsows the an It method exemplary is believed of det image thatermining it of should a vonthe be parame-Karman in the tersrangestreet l and captured from h. 0.15The with todistance 0.45 a camera. [22 l ].is Figuredetermined The 19image shows from shows an the exemplary the canter method of imagethe of shedderdet ofermining a von bar Karman to the the parame- canter street ofcapturedters the l and vortex withh. The image. a camera.distance The distance Thel is determined image h is shows the from di thestance the method canter from of theof determining the flow shedder axis to thebar the parametersto edge the canterof thel vortex.andof theh. vortex The distance image.l Theis determined distance h from is the the di canterstance offrom the the shedder flow baraxis to to the the canter edge of thethe vortexvortex. image. The distance h is the distance from the ﬂow axis to the edge of the vortex.

l l

h h

Figure 19. Schematic of geometry of von Karman vortex street. Figure 19.19. Schematic of geometry of vonvon KarmanKarman vortexvortex street.street. The next part of the research is to determine the relationship between the characteristic dimensionThe nextnext partpart of of theof thethe vortex research research shedder is is to to determine bardetermine d and the thethe relationship dimensionsrelationship between characterizingbetween the the characteristic character- the von Karmandimensionistic dimension vortex of the street.of vortex the Moreover,vortex shedder shedder bar it wasd and bar determined thed and dimensions the how dimensions the characterizing change characterizing of the vonshape Karmanthe of vonthe vortexKarman street.generator vortex Moreover, street. influences Moreover, it was the determined change it was of how determinedthe ch thearacteristic change how of thethedimensions shapechange of theofof the vortex vortexshape generator ofstreet the (Figureinfluencesvortex generator 20). the change influences of the characteristicthe change of dimensionsthe characteristic of the vortexdimensions street of (Figure the vortex 20). street (Figure 20).

Figure 20. InfluenceInﬂuence of shedder size on ( a) length of the vortex zone l (b) distance from axisaxis hh.. Figure 20. Influence of shedder size on (a) length of the vortex zone l (b) distance from axis h. These equations can be used to calculate the dimensions characteristic for paths resulting from the ﬂow of vortex shedder bar of other shapes.

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Subsequent research consisted in determining the effect of changing the shape of the vortex shedder on the characteristic dimensions of the von Karman vortex street (Table4 ). The d dimension of all tested shedder bars was the same and amounted to d = 30 mm. The results were compared with a cylindrical shedder with a diameter of d = 30 mm. For this shedder, the characteristic dimensions of the von Karman vortex street were ho = 18.2 mm and lo = 51.5 mm. The results presented in Table3 are averaged values for the liquid velocity from 0.3 m/s to 1 m/s.

Table 4. Inﬂuence of changing the shape of the vortex shedder bar on the average dimensions of the vortex street.

Shedder Shape h [mm] l [mm] h/l h/ho l/lo h/d l/d E1 18.75 47.9 0.39 1.03 0.93 0.63 1.6 E2 10.38 52 0.2 0.57 1.01 0.35 1.73 E3 16.9 42.23 0.4 0.93 0.93 0.56 1.41 E4 10.37 49.44 0.2 0.57 0.96 0.34 1.65 E5 10.56 47.38 0.22 0.58 0.92 0.35 1.57

As it results from the research, changing the shape of the rake surface to a more streamlined one shortens the non-vortex zone behind the shedder (E1). In turn, the introduction of a slot on the lower and upper surface of the generator causes a slight elongation of this zone (E2). The introduction of the slot, in turn, signiﬁcantly reduces the distance h. Modiﬁcation of the generator runoff surface to be more streamlined causes that the distance h is greater (E5). Good stability of the swirl street while maintaining favorable values of h and l is obtained for the E 4 shape of the vortex shedder bar. The shape E3 reduces the characteristic dimensions of the vortex street by 7% compared to the cyclic shedder, but introduces the greatest instability of the resulting vortices (h/l = 0.4). The introduction of a sharp rake face usually increases the distance h.

6. Conclusions The paper presents results of numerical studies on the impact of changes in surface of flow around vortex shedder on the parameters of von Karman vortex street. Based on the numerical results, several shapes of vortex shedders were selected, which improve the metrological parameters of the vortex flow meters (Figure3). Selected shapes were subjected to experimental tests on a test stand. In this way, the results of numerical tests were verified and it was found that the most useful solution from the proposed solutions of shedders bars is the E4 shape for metrological reasons. For the purposes of the study, a method of visualization of the von Karman vortex street was developed. The measurement method presented in the work can be used to determine the velocity of liquid flow in the visualization processes. The only limitation is the use of trans-parent liquids. Due to the fact that many transparent liquids exist in technical conditions, its range of use is very large. The presented approach offers the possibility to determine liquid flow velocity with a precision in the range of up to a few per cent in relation to the velocity of the liquid flow. The uncertainty of such measurements is considerably relative to the liquid velocity. An adequate selection of the exposition time forms a vital aspect in ensuring the required accuracy of the measurements; yet it is relative to the velocity of the analyzed phenomenon. The analysis described in this paper provides valuable insights into the selection of exposition times in studies involving visualization of the liquid flow. The measurement method can be used to visualization the velocity of liquid flow in the other processes. The only limitation is the use of trans-parent liquids. Due to the fact that many transparent liquids exist in technical conditions, its range of use is very large. The presented approach offers the possibility to determine liquid flow velocity with a precision in the range of up to a few per cent in relation to the velocity of the liquid flow. Sensors 2021, 21, 4697 19 of 20

This paper presents into the selection of exposition times in studies involving visualization of the liquid flow, which has a great influence on the accuracy of the measurement. On the basis of the observations, it was found that the introduction of the fracture improves the stabilization of the vortex street. Jet is beneficial in terms of improving the regularity of vortex formation. When the rake surface is more streamlined, it helps to stabilize the stream at higher speeds of movement. W guiding the sharp edges on the trailing surface causes the distance h to be greater, but improves the widening of the lower range of vortex formation. The introduction of the concave rake face destabilizes the vortex street, which is a disadvantage.

Author Contributions: Conceptualization, M.R.R. and B.C.-N.; methodology, M.R.R.; validation, B.C.-N.; formal analysis, M.R.R.; writing—original draft preparation, B.C.-N.; writing—review and editing, M.R.R.; supervision, M.R.R.; All authors have read and agreed to the published version of the manuscript. Funding: Opole University of Technology. Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Institutional Review Board. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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